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INVARIANT  TWINS
floating, like an atmosphere, around it - but infinite as were
these derived existences, these emanations from the parent
form, it is found that they admit of being obtained by composi-
tion, by mixture, so to say, of a certain limited number of
fundamental forms, standard rays, as they might be termed in
the Algebraic Spectrum of the Quantic to which they belong.
And, as it is a leading pursuit of the Physicists of the present
day [1877, and even to-day] to ascertain the fixed lines in the
spectrum of every chemical substance, so it is the aim and
object of a great school of mathematicians to make out the
fundamental derived forms, the Covariants [that kind of 'inva-
riant' expression, already described, which involves both the
variables and the coefficients of the form or quantic] and
Invariants9 as they are called, of these Quantics,'
To mathematical readers it will be evident that Sylvester is
here giving a very beautiful analogy for the fundamental
system and the syzygies for a given form; the non-mathematical
reader may be recommended to re-read the passage to catch the
spirit of the algebra Sylvester is talking about, as the analogy
is really a close one and as fine an example of 'popularized'
mathematics as one is likely to find in a year's marching.
In a footnote Sylvester presently remarks *I have at present a
class of from eight to ten students attending my lectures on the
Modern Higher Algebra. One of them, a young engineer,
engaged from eight in the morning to six at night in the duties
of his office, with an interval of an hour and a half for his dinner
or lectures, has furnished me with the best proof, and the best
expressed, I have ever seen of what I call [a certain theorem].
-. .* Sylvester's enthusiasm  he was past sixty - was that of a
prophet inspiring others to see the promised land which he had
discovered or was about to discover. Here was teaching at its
best, at the only level, in fact, which justifies advanced teaching
at all.
He had complimentary things to say (in footnotes) about the
country of his adoption: *... I believe there is no nation in the
world where ability with character counts for so much, and the
mere possession of wealth (in spite of all that we hear about the
Almighty dollar), for so little as in America. ...'
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